The principle whereon the majority of the mechanical systems are based, which are nowadays in use for the measurement of the absolute verical displacement is that of the measurement of the relative displacement with respect to an ideally inertial mass. The existing realizations integrate mechanical principles and techniques of various nature and accuracy [1] [2] [3].
The quality of the realization of the reference inertial mass limits the sensitivity of the system in general, and most of all the frequency band of the system, in a particular way in the region of low frequencies. This notwithstanding, even in the hypothesis one had at disposal a really inertial mass, the measurements would be in any case limited by the thermal noise of the same mass, the noise of the reading system and the interaction of the sensor with the ambient noises [1] [2] [3].
In the case of vertical displacement measurements, such systems in general make use of or can be related to an oscillating system, e.g. a strap of suitable length locked at one of its ends. On the other end of such a tape a suitable mass is positioned, which constitutes as a matter of fact he inertial mass. The realization of a measurement with good sensitivity at low frequencies implies, therefore, that the mechanical oscillating system has very low resonance frequency together with a good mechanical quality factor. Generally this system is realized by means of a feed-back system, so that the inertial mass is maintained fixed in the position chosen as reference by means of a feedback control system. The error signal of the control system, as obtained by suitable measurement sensors, provides the vertical acceleration signal, wherefrom it is possible to extract the absolute vertical displacement signal, obviously limited by the sensitivity and the measurement band of the system.
The following are of utmost importance in the realization of a mechanical system for the measurement of the absolute as well: an effective decoupling of the vertical degree of freedom from the other degrees of freedom (horizontal movements, rotations, etc.) and a high mechanical quality factor, which is an index of reduced energetic leakages of the oscillating system (thermal noise of the joints, viscous effect of the air, etc.), necessary for the apparent movement of the inertial mass not to be influenced by the movement of the fixed part of the mechanical system supporting such a mass.
Mechanical systems a with very low resonance frequency generally have, however, very large dimensions, are complex to be realized and calibrated, and most of all they are not dimensionally scalable. Indeed, such systems require firstly the balancing of the gravity force acting on the inertial mass, which, especially at low frequencies, becomes a problem owing to the weight of the same mass, and, therefore, the necessity of large forces to be applied for its positioning. Moreover, as one widen the measurement band of these low-frequency sensors the system sensitivity to the ambient noises widely increases (variations of temperature, pressure, humidity, etc.) as well as the problems of decoupling between the vertical degree of freedom and the other degrees of freedoms [1] [2][3].
Only the low-frequency system constituted by a Watts pendulum [4] differs from all the existing mechanical systems, which is realized also in the monolithic form (with joints worked by electro-erosion), both in the classically known experimental embodiment with joints in traction [5] [6], and in a new embodiment with some joints in compression (Italian patent application [7], here integrally included by reference), which unites, to a full dimensional scalability of the sensor, a full tunability of the resonance frequency, high quality factors, large measurement frequency band, reduced problems of coupling between the various degrees of freedom and wide insensitivity to the ambient noises, as a consequence of an efficient signal readout system based on optoelectronic methods, for example optical levers and laser interferometers [8]. The progresses in the technological development of such a folded pendulum, which however owing to its structure can be used only for horizontal applications, are described by a wide literature [9-17].